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THE general introduction in the first volume of this book need
not be repeated here. However, a few points may be briefly
recalled, as they are relevant for both volumes. We shall also
take a quick glance ahead beyond the last of the men mentioned.
Mathematics as understood by mathematicians is based on
deductive reasoning applied to sets of outright assumptions
called axioms or postulates. It is sufficient here to describe
deductive reasoning as the rules of common logic, although
mathematical logic goes far beyond that. The postulates under-
lying a particular division of mathematics, such as elementary
algebra or school geometry, may have been suggested by every-
day observation of the world as it presents itself to our senses.
Many of the propositions of geometry, for instance, such as that
gem attributed to Thales in the sixth century B.C., "The angle
inscribed hi a semicircle is a right angle1, are evident to the eye.
But however obvious and sensible they may seem, they are not
a part of mathematics until they have been deduced from a set
of postulates accepted without argument as self-consistent. The
great but (to us) nameless mathematicians of Babylonia dis-
covered, or invented, many beautiful things in both algebra and
geometry, but, so far as is known, t&ey proved none of them.
It remained for the Greeks of about 600 B.C. to invent proof -
deductive reasoning. With that epochal invention mathematics
was born. But logic and proof are by no means the whole story.
Intuition and insight are as freely used to-day in mathematics
as they must have been by the Babylonians.
It was many centuries before the full significance of what
those old Greeks had done was understood and applied to all
mathematics, and thence to all reasoning. A notable instance is
school algebra, first thoroughly understood and rigorously
developed only in the 1830's by the British School, of whom
George Peacock (1791-1858) is especially memorable. Unfortu-
nately there is not space here to tell the lives of these little-